Don’t bother with a calculator…
Hi reader! With this article, I intend to have you walk away with the logic skills to figure out simple math problems in your head, without a calculator. I am going to do my best to explain the logic techniques, but once you begin to understand the basic idea, the rest will be easy.
Introduction
So, let’s start with the overall concept first before I offer specific strategies for different types of problems. Basically, the idea is to break the big problem we’re trying to do into super-easy steps. Here’s an easy example:
Say you are browsing items in a store and you come across a couple of items with different prices. How much are these items going to cost together?
$1.50 + $17.37 = ?
We want to break this into smaller steps. When dealing with decimals (like money) it’s easiest to start with the change before adding the bills. Let’s take out the two amounts of change and work with that.
$0.50 + $0.37 + $1.00 + $17.00
This is the exact same thing as our original problem just expressed differently. So let’s try simplifying it even further.
$0.87 + $18.00
$0.87 + $18.00 = $18.37
It may seem a little intimidating at first, but it really isn’t too hard. At first glance, our original problem seemed a little overwhelming, but as you can see the smaller steps can easily be done in your head. Just add the cents, then the dollars, and combine the answers.
I know this isn’t homework and there’s little incentive, but try these problems in your head for practice. After going through it a few times on your own, it will start to become easier.
1. $2.05 + $0.25
2. $1.00 + $3.99
3. $23.44 + $10.12
4. $49.21 + $57.35
Dealing with $0.99
I hope that so far I am explaining things well. A better real-world example might be to include two (or more) things with 0.99 tacked to the end. Let’s start with a second example.
$0.99 + $0.99
A little more annoying, right? Well, try thinking about this way: $0.99 may as well be a dollar. But, because of that $0.01 cent we’re trimming off, that extra cent needs to be compensated for. Let’s try expressing it a little differently.
$1.00 + $1.00 - $0.01 + $0.01
$2.00 - $0.02
$2.00 - $0.02 = $1.98
I find this to be the easiest way to view the problem. Remember, we decided to just see each $0.99 as a dollar. But, also remember that for each $0.99 that we round up, we are gipping ourselves out of that extra cent. In order to make things right all we had to do was take $0.01 off the total for each $0.99 we round up. So if we bought 5 packs of gummy worms for $0.99 a piece, that may as well be $5.00. But, since we rounded up the price for each pack of worms, we’re screwing ourselves out of $0.05. To make things right, we basically give ourselves a five cent coupon off the total.
Once we’ve dealt with the $0.99 cents issue, we can add the dollar amounts. Then we combine the dollar total with the total we got from the $0.99 problem.
$5.99 + $9.99
$5.00 + $9.00 + $0.99 + $0.99
$5.00 + $9.00 + $1.00 + $1.00 - $0.02
$5.00 + $9.00 + $2.00 - $0.02
$5.00 + $9.00 + $1.98
$14.00 + $1.98
$15.98
This same rounding up technique can be applied to any amount of cents, just remember to take back the difference that was skipped to get up to a dollar (i.e. rounding $0.60 to a dollar means you need to remember to take that extra $0.40 off the total).
Here are a few problems to try.
1. $0.99 + $0.99
2. $20.99 + $13.99
3. $99.99 + $0.99
4. $5.35 + $0.99
Subtraction
Subtraction can be a little trickier to do in your head. But don’t worry; there are tricks to subtraction as well.
Let’s start with simple subtraction, void of any decimals.
40 – 17 = ?
I’ve found a logical technique to make it a little easier to do. Subtracting clean, even numbers is ideal. So, if we deal with the pesky 7 (or any non-zero number) first, subtracting an even number ending in zero is easy.
40 – 7 – 10 = ?
33 – 10 = 23
Now that we’ve dealt with the 7, subtracting 10 from 33 is simple without a calculator. Keep in mind though that’s we do not need to alter the number we’re subtracting from (in this case the 40), despite whether or not it ends in a 0. The only number that needs to be simplified is the amount we’re taking away. Let’s go through another example.
39 – 25 = ?
Remember, the first thing that should be done is to just do subtraction between the 39 and the ‘5’ from 25.
39 – 5 – 20 = ?
Doing that would leave us with…
34 – 20 = ?
The rest of the problem should end up being easy enough to do on your own. 34 minus 20 is 14.
34 – 20 = 14
Transferring this logic over to decimals is smooth, since we deal with the decimal part before the whole number part. Here’s an example.
12.46 – 9.42 = ?
Remember that when we deal with decimals (like money) it’s easier to deal with the decimal part first.
12.00 – 9.00 + 46 – 42
12.00 – 9.00 + 46 – 2 – 40
12.00 – 9.00 + 44 – 40
12.00 – 9.00 + 04
3.00 + 04
3.04
Here are some practice problems.
1. 93 – 21
2. 77 – 51
3. $29.99 - $10.50
4. $120.23 - $10.16
When an answer is beyond zero
I’m sure you know that when working with whole integers you start hitting the negatives if the amount you are taking away is bigger than the number you are subtracting from. With decimals however, that’s not the case. If you have $25.50 and you are about to spend $0.60 on something, you aren’t going to have negative cents. Like all the other math we’ve covered, the problem can be worked through with logic and by trying to see the problem from an angled perspective. This is no different.
The way I tend to look at these kinds of problems is like this: when I hit zero, I immediately pause my subtraction and deduct myself a dollar, then continue my subtraction, but from 100 (100 cents to a dollar). Here’s an example.
$10.50 - $0.60 = ?
If we go through this in slow motion, it may be easier to understand.
So, we have begun subtracting our $0.60 from $0.50 (remember, we evaluate our decimals first, for this exact reason).
$10.50 - $0.60
$10.40 - $0.50
$10.30 - $0.40
$10.20 - $0.30
$10.10 - $0.20
$10.00 - $0.10
We have now reached zero and used up this dollar, but still have $0.10 to go. Because we’ve officially used up a dollar, we drop to the next dollar from our $10.00.
$10.00 - $0.10
$9.90
Now that we have compensated for our spent dollar, we can resume our subtraction starting from the 100 from our fresh dollar.
Practice problems:
1. $24.99 - $20.00
2. $77.30 - $11.50
3. $5.30 - $2.85
Multiplication
Multiplication is actually pretty easy to do. The basic principle of multiplying without a calculator is to basically keep multiplying by smaller amount until you’ve reached the right number of multiples. Look at the following example.
13 x 8
Ok, so we know that multiplying something is equivalent to adding that something over and over. We can take advantage of that for easy multiplication. If you can double a number in your head you can solve most multiplication problems. Let’s try to solve the example.
13 x 8 = 13 x 2 x 2 x 2
13 x 2 = 26
26 x 2 = 52
52 x 2 = 104
13 x 8 = 104
Square Roots
This one doesn’t involve much logic; it’s more of a guess and check system and the answer most likely won’t be exact. The main idea is to figure out what numbers the answer is between. Let’s try an example. (Sq(number) = square root)
Sq(11) = ?
First, we find what the answer to an easier square root problem is both below and above 11. So an easier square root below 11 might be 9. An easy problem above 11 is 16.
Sq(9) = 3
Sq(16) = 4
So, because Sq(11) is between Sq(9) and Sq(16), then we know that the answer to Sq(11) is between 3 and 4. A good guess would be 3.5. You COULD narrow it down further by seeing if Sq(11) is closer to Sq(9) or Sq(16). If it’s closer to the lower one, then the answer would be between 3.0 and 3.5 and between 3.5 and 4.0 if the problem is closer to the higher one. The answer would be around 3.5 if Sq(11) would be right in the middle between 9 and 16.
Obviously Sq(11) is closest to Sq(9), so a good guess might be 3.2 or 3.3. If you did this on a calculator, the answer would be 3.316.
If you have any question, please email me at [email protected]. I hope this guide helped.
